Random Planar Lattices and Integrated SuperBrownian Excursion

De
Chessel

6
pages

Random Planar Lattices and Integrated SuperBrownian Excursion Philippe Chassaing, Gilles Schae?er ABSTRACT: In this extended abstract, a surprising connection is de- scribed between a speciﬁc brand of random lattices, namely planar quadrangula- tions, and Aldous' Integrated SuperBrownian Excursion (ISE). As a consequence, the radius rn of a random quadrangulation with n faces is shown to converge, up to scaling, to the width r = R ? L of the support of the one-dimensional ISE, or more precisely: n?1/4rn law?? (8/9)1/4 r. The combinatorial ingredients are an encoding by well labelled trees, reminiscent of the work of Cori and Vauquelin, and the conjugation of tree principle, used to relate the latter trees to embedded (discrete) plane trees in the sense of Aldous. From probability, we need a new result of independent interest, namely the weak convergence of the encoding of a random embedded plane tree by two contour walks (e(n), W (n)) to the Brownian snake description (e, W ) of ISE. 1 Introduction From a distant perspective, this article uncovers a surprising, and hopefully deep, relation between two famous models: random planar maps, as studied in combi- natorics and mathematical quantum physics, and Brownian snakes, as studied in probability and mathematical statistical physics.